Sample spaces and counting: outcomes, order and equal likelihood

Define the experiment before counting

Imagine four distinct labels, A, B, C and D, and a procedure that chooses two different labels. The statement still leaves an important question unanswered: does the order of selection matter? Choosing A then B is a different recorded outcome from choosing B then A if the record contains first and second positions. If the result is simply a two-member group, those descriptions identify the same outcome. The Stanford probability refresher defines sample spaces and events; the practical task here is to make those definitions precise enough for another person to repeat the experiment.

Use multiplication for successive choices

In the ordered experiment without replacement, there are four possible first labels and three possible second labels after any first choice. Multiplying gives twelve ordered outcomes. The Penn State counting lesson distinguishes ordered arrangements from selections that ignore order. Multiplication here counts available branches; it does not itself prove those branches have equal probabilities. If a mechanism favours A, the same twelve outcomes remain possible but they need not carry identical weights. Keep the inventory of possibilities separate from the probability assigned to each possibility.

Compare four versions of the experiment

The four rows answer different questions. A larger count is not a more accurate answer unless it corresponds to the specified experiment. For the unordered pair {A,B}, the two ordered records A,B and B,A are merged once, which explains the division by two without memorising a formula.

Turn a collection of outcomes into an event

In the ordered, no-replacement version, let E mean that A appears somewhere. Six outcomes satisfy E: A can occupy the first position with three partners, or the second position with three partners. Under a uniform selection mechanism, P(E)=6/12=1/2. An event is a set of outcomes, so it can contain several sequences even when described by one sentence. Write a short list for a small example before generalising. That list often reveals an omitted case, a duplicated case or a hidden restriction that a formula alone would conceal.

Check complements and overlapping descriptions

The complement of E contains the ordered pairs drawn from B, C and D, giving three times two equals six outcomes. This independently checks the result. If another event F means that B appears, E and F overlap on A,B and B,A. Adding their probabilities without subtracting the overlap counts those two records twice. OpenStax’s treatment of events helps distinguish overlapping sets, mutually exclusive events and independence. These are separate properties: two labels can appear together, while the mechanism still makes their appearances dependent.

Know when counting alone is insufficient

For a continuous measurement such as a hypothetical waiting time, there may be infinitely many possible values. Dividing the number of favourable values by the number of possible values is then not a workable counting rule. Even in a finite space, equal likelihood must come from a model or supported mechanism rather than from the neatness of a list. A useful report states both components: the possible outcomes and their probability weights. When the weights are unknown, an exact count can still be correct while a numerical probability remains unjustified.

Explain why repeated items change the count

If labels are no longer distinct, the counting object changes again. Four physical tokens marked A, A, B and C can be distinguished by their physical identities even though two display the same letter. An experiment recording token identities has a different sample space from one recording only printed letters. State which observation is retained before counting outcomes. Merging records can also merge unequal numbers of underlying possibilities, so equal likelihood at the physical-token level need not become equal likelihood among the displayed letter combinations.

Counting checklist and a verification habit

• State the recorded outcome precisely.
• Decide whether order matters.
• Specify replacement and repetition.
• Count a small example explicitly.
• Justify any equal-likelihood assumption.
• Check overlaps and complements.
• Keep counts distinct from probabilities.

For an expanded version with five labels, repeat the reasoning rather than copying the old denominator. Ordered distinct pairs become five times four, while unordered pairs divide that total by two. A spreadsheet or short enumeration can confirm the count, but its output is only as sound as the rule used to generate records. Document the rule with the result so a later reader can identify exactly what was counted.

Sources and verification

• Probability and statistics refresher — Stanford University teaching materials — 2026-10-23
• Counting techniques, STAT 414 — Penn State — 2026-10-23
• Independent and mutually exclusive events — OpenStax, Rice University — 2026-10-23